UMBC CMSC441, Design & Analysis of Algorithms, Fall 2001, Section 0101

Homework 1, Due Tuesday September 11, 72 points total

1. (12 points each) Prove that the following statements are true:

   1. 4n³ - 6n = Theta(n³)

   2. n! = O(nⁿ)

   3. n^1.2 + n lg n = Theta(n^1.2)
    

2. (12 points each) Show that the following statements are false:

   1. 3 n² -9n = O(n)

   2. n² lg n = Theta(n²)
    

3. (12 points) Prove by induction that for x != 1 and n >= 0 that

   (Summation of xⁱ for i=0 to n) = (xⁿ⁺¹ - 1)/(x-1).